Problem: Suppose you measure three numbers as follows: x = 200§2; y = 50§2; z = 40§2; where the three uncertainties are independent and random. Use step-by-step propagation to find the quantity q = x=(y ¡ z) with its uncertainty. Solution: Let D = y¡z = 10§2 p 2 = 10§3. Then q = x D = 20§20 p 0:012 +0:32 = 20§6: 10/5/01 7

(ii) Solved Example Problem for Error in the difference of two quantities. Example 1.6. The temperatures of two bodies measured by a thermometer are t 1 = (20 + 0.5)°C, t 2 = (50 ± 0.5)°C. Calculate the temperature difference and the error therein. Solution

Several formulas were presented for propagating random errors through calculations using partial derivatives from calculus. The formulas assume a normal distribution of random errors and no correlation between errors. The simplified formulas are summarized below. For complicated

Probable error: The symmetric range about the mean, within which there is 50% chance that a measurement will fall. The width of the range depends on the distribution of the variable. For example, for Gaussian distributed error, the probable error is 0.675 .

In order for us to determine what happens to the uncertainty (error) in the length of the rod or volume of the block we must analyze how the error (uncertainty) propagates when we do the calculation. In error analysis we refer to this as error propagation.

There are two kinds of error that one deals with in the lab: random error and systemic error. Random error, per the name, are statistical e ects that push your data every which way from the \true" result, with no consistency to its e ects. These will result in your data being right on average, but any individual data point will be o . This ...

Calculate (i) the mean value of the period of oscillation (ii) the absolute error in each measurement (iii) the mean absolute error (iv) the relative error (v) the percentage error. Express the result in proper form.

USING DIFFERENTIAL CALCULUS TO DETERMINE THE PROPAGATION OF ERRORS. Suppose that an experiment is performed in which only two physical quantities (x, y) are measured so that the desired result quantity can be computed, R(x, y).

Random errors can occur through a variety of factors and do not follow a pattern such as the systematic errors. That means random errors corrupt your readings in a completely random and unpredictable fashion. Random errors can be averaged out by taking a large number of readings.

Example, if x = w 1/w 2, (or w 1 ·w 2) σ x x = s σ2 w1 w2 1 + σ2 w2 w2 2, i.e. fractional errors add in quadrature. Having made N measurements we quote our best (max-imum likelihood) values as: Distribution Mean hxi = 1 N X x i ≈ µ = P (x i/σ i)2 P (1/σ i)2 (10) Distribution Uncertainty σ x = σ one measurement√ N = r N N −1 sP (x ...